Related papers: How many ways can you make change: Some easy proof…
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
We show that exchangeable structures such as Polya urn model and Chinese restaurant process can be constructed from sets with a real number of elements. From this construction, the exchangeability of these structures becomes obvious and the…
It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually…
A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely…
People often engage in costly cooperation, especially in repeated interactions. When deciding whether to cooperate, individuals typically take into account how others have acted in the past. For instance, when one person is deciding whether…
This article is about the mathematics of ringing the changes. We describe the mathematics which arises from a real-world activity, that of ringing the changes on bells. We present Rankin's solution of one of the famous old problems in the…
Consider the following game: You are given two indistinguishable envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope, you are given…
Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broad…
Let $q \in (0,1)$ and $\delta \in (0,1)$ be real numbers, and let $C$ be a coin that comes up heads with an unknown probability $p$, such that $p \neq q$. We present an algorithm that, on input $C$, $q$, and $\delta$, decides, with…
The quanto option is a cross-currency derivative in which the pay-off is given in foreign currency and then converted to domestic currency, through a constant exchange rate, used for the conversion and determined at contract inception.…
Let $a,b$ be positive, relatively prime, integers. We prove, using induction, that for every $d > ab-a-b$ there exist $x,y\in\mathbb{Z}_{\geq 0}$, such that $d=ax+by$. As a byproduct, we obtain a constructive recursive algorithm for…
This paper shows that, for matrix multiplications and convolutions, it is possible to asymptotically replace each real multiplication with a single squaring operation. Similarly, a single complex multiplication can be replaced with 3…
Quantum money is a cryptographic protocol in which a mint can produce a quantum state, no one else can copy the state, and anyone (with a quantum computer) can verify that the state came from the mint. We present a concrete quantum money…
We study the computational complexity of converting one representation of real numbers into another representation. Typical examples of representations are Cauchy sequences, base-10 expansions, Dedekind cuts and continued fractions.
I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews' $q$-Dyson conjecture.
We introduce a new family of one-player games, involving the movement of coins from one configuration to another. Moves are restricted so that a coin can be placed only in a position that is adjacent to at least two other coins. The goal of…
Given $n$ real numbers $0\leq x_1,...,x_n<1$ and a permutation~$\sigma$ of $\{1,...,n\}$, we can always find $\xbar_1,...,\xbar_n\in\{0,1\}$ so that the partial sums $\xbar_1+... +\xbar_k$ and $\xbar_{\sigma 1}+... +\xbar_{\sigma k}$ differ…
We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…
We consider the problem of factoring permutations as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances. In particular, we investigate the minimum number, $\delta$, such…
A series of examples of computational models is provided, where the model aim is to interpret numerical results in terms of internal states of agents minds. Two opposite strategies or research can be distinguished in the literature. First…